stochastic filtering - a brief tutorialrmcastro/publications/stochastic_filtering.pdf · stochastic...

Stochastic FilteringStochastic Filtering

A brief tutorial

Rui Castro

© 2006 - Rui Castro

MotivationIn many applications one wants to track time-varying processes

Example: tracking temperature in a museum room with a inaccurate thermometer with large error.

How can we accurately estimate the room temperature???


Motivation

Some kind of windowed estimate should do the trick... (low pass filtering)

Key fact: Knowledge of the process dynamics (i.e., temperature changes sloooooowly...


Modeling - Dynamics

The distribution of depends only on the value of .Together with the knowledge of this completely defines the random process

Dynamics Kernel

This random process is the quantity we are interested in estimating (e.g., temperature of the room)


Dynamics - Example

Position-Velocity (PV) Model:

Discrete counterpart:

Velocity is modeled as Brownian motion


Observation Model

In general we cannot observe the process directly

Model Assumptions:

• Observation at time n depends only on state at that time

• Observations do not affect the physical process

Observation Likelihood


Main Goal

Given a set of observations up to time n,

we want to estimate .

We follow a Bayesian approach, therefore all the information about is provided by the density

Filtering Density

How do we do it???© 2006 - Rui Castro

Can we estimate ?

After we observe we can use Bayes rule

Density Propagation – First Observation

Prior Density


Density Propagation – Prediction

Can we predict ?


Our prediction is then given by

Prediction Density



Density Propagation – General

Start with a prediction based on previous observations

Prediction Density

After we observe we can compute


Density Propagation – General

We can now predict


Density Propagation – Graphically


Density Propagation – Filtering


Main Goal

Now that we have

what do we do with it? That is, how do we construct an estimate

Many possibilities, for example if we want to minimize

the optimal choice is


A Particular Case: The Kalman Filter

And, very important

In other words, we assume that



Since

are all Gaussian it follows that

and

are also Gaussian (because the product of two Gaussians is a Gaussian and the convolution of two Gaussians is also a Gaussian).

Therefore we need only to propagate means and covariance matrices (a nice closed form solution)



Start with the prediction density

After we observe compute

Compute the updated prediction



Taking

yields the Kalman Filter

Numerous applications, but restricted to linear models

What if models are non-linear???


The Extended Kalman Filter

Suppose is normal

If our prediction density is “accurate” then

Idea: Linearize around the prediction mean



Similarly, if



This procedure is widely used in practice, for example in the

Global Positioning System – GPS

The Extended Kalman filter works is you have an accurate clock and know where you are... If not the linearization is poor and there’s not much you can do

Distance measured using an

accurate clock


Other Non-Linear Model Procedures

Approximation of non-linearities by Gaussian Mixture models

Evaluate the densities only at a finite (fixed) set of points – the Point-Mass Filter

Effective only in small state spaces!!!© 2006 - Rui Castro

Particle Filters

Same idea as in the point-mass filter, but dynamically allocate the location of the point masses.

In other words, describe the densities in a empirical way, though a set of samples:

The samples can be used to compute almost

anything interesting about the density


Condensation Algorithm (Blake et al)

Start with samples from

Importance Sampled

Filtering Density


Condensation Algorithm

Sample according to these importance samples and simulate the dynamics model

Keep repeating these two steps as you collect more observations


Condensation Algorithm

This procedure is asymptotically optimal as the number of particles grows

Many “tricks of the trade” can be used to improve performance and reduce computational cost, namely:

• Choose the number of particles dynamically• Augment/reduce the size of the dynamical model to allow multi-target tracking

Key Fact: For computationally feasible procedures the model of the dynamics must be easily simulated.

Recently, non-asymptotic guarantees for these procedures became available.


A Tracking Example

Tracking an orange (clementine), in clutter (including other oranges). Movie taken with a handheld camera (no camera motion registration).


A Tracking Example

A particle filter tracking the left-most orange, using a PV dynamics model and 1000 particles.


Final Remarks

Stochastic Filtering is a very general (Bayesian) framework for sequential estimation in a model-based setting.

For linear and Gaussian models the densities being propagated have a closed-form solution and the result is simply the well known Kalman filter.

When using non-linear models closed-form solutions are generally not possible, and so approximate methods need to be used.


stochastic filtering - a brief tutorialrmcastro/publications/stochastic_filtering.pdf · stochastic...

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